When I am trying to connect to Oracle 10g with the connection string, it works fine but when I am trying to connect same Classic asp code with Oracle 11g, I am getting below error.

Microsoft OLE DB Provider for ODBC error '80004005'

[Microsoft][ODBC Driver Manager] Data source name not found and no default driver specified.

I have checked my TNS entry is correct under tnsnames.ora

I am using Windows server 2008 and Oracle 11g.

The Microsoft ODBC driver for Oracle is very old and not supported.The support for Oracle database and drivers is at Oracles Web Site.
I suggest you visit oracle.com/technetwork/database/windows/whatsnew/index.html Oracle Network and register for an Account which is no cost.Then after you have joined the Oracle Network check the download sectio for Visual Studio Tools for Oracle and look for the ODAC112021xcopy.zip package for X64 or X86 which contains the updated runtime drivers for Windows.
After you have downloaded the runtime drivers read the guides at oracle for install.The new driver for Oracle will appear in the ODBC manager you then can update "your connection string" to use the new driver.If you can not connnect to the Oracle database the problem typically are the tns.listener connection settings.
Typical settings use XE.NAME for the server instance. You will need to open the firewall for port 1521 to allow connections.
If you look at the ora error which is oraXXXX message and search (Goolge or Bing) there are place on the Web that show how to change the tns listener for the Oracle Database engine.
If you have problems with the connection string you can (Google or Bing) Connection Strings.
If you have IIS Server errors then read the IIS Net library guides for Enabling and Turning on "Failed Request Tracing"